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Magnetic field dependent magnetization oscillations in two-dimensional quantum electron systems
Volume 129, number 2
PHYSICS LETTERS A
9 May 1988
MAGNETIC FIELD DEPENDENT MAGNETIZATION OSCILLATIONS IN TWO-DIMENSIONAL QUANTUM ELECTRON SYSTEMS
J.P. PENG Department ofphysics, Fudan University, Shanghai, PR China
Received 13 July 1987; accepted for publication 1 March 1988 Communicated by A.A. Maradudin
A simple relationship between amplitudes of magnetization oscillations is presented as result of a single-electron Josephson junction in a two-dimensional quantum electron system with a perpendicular magnetic field.
Recent high-precision dc measurements of two-dimensional electron systems (2DES) in homogeneous high-mobility GaAs/AlGaAs single-layer and multilayer heterostructures [ 1 ] as well as in single crystals of ( TMTSF)&104 in the regime where the magnetic field induces a series of transitions show interesting pictures [ 21. Anisotropic oscillation of the magnetization has been observed whose amplitude attenuates with decreasing magnetic field and shows no observable spin splitting. In GaAs/AlGaAs heterostructures well-developed quantum Hall effects have been observed at low temperatures and in (TMTSF ) Z,ClO, single-electron crystals Shubnikov-de Haas (SdH ) type magnetoresistance oscillation as well as step-like Hall resistances have been observed [ 3 1. It is also worth mentioning that small, smooth isotropic and large anisotropic background magnetizations have been observed in these samples. Although the magnetoresistance and Hall data resemble SdH besides that the magnetization results resemble de Haas-van Alphen (dHvA) oscillations, there are strong reasons to believe that the dHvA oscillation gives not the entire picture of the magnetization. For a 2DES with fixed particle density, such oscillations at zero temperature appear as discontinuous diamagnetic (as the field is increased) jumps of magnitude &n/m* (field independent), where n is the zero-field carrier density, with a paramagnetic relaxation between jumps [ 41. Also in the regime where few Landau levels are occupied, there should be an observable spin splitting. For a fixed chemical
potential the jumps are paramagnetic and the recovery is diamagnetic and the curves will appear inverted. As the temperature is increased, the discontinuities are “smeared” but the magnetic fields at which they occur should remain the same. But even if only a small smearing fron the zero-temperature behavior is assumed, there are still major discrepancies related to the increasing amplitude with applied magnetic field, the apparent absence of spin splitting in the case of (TMTSF) &lo, single crystal. In this Letter a quantitative description is presented for the oscillatory amplitude attenuation with increasing applied magnetic field in quantum 2DES as well as a comparison with recent experiment. Motivated by the concept of frustration in two-dimensional superconducting arrays, a theory of the quantum Hall effect has been given in the preceding Letter [ 51 with the hypothesis of the single-electron Josephson arrays where quantization of the London flux and the formation of Cooper pairs are essential [ 5 1. The London quantization condition in a cell is s
(mv/e+A)dL=h/e,
(1)
which can be written as
s
(Ai+A)*dL=h/e,
(2)
where A is the vector potential of the applied magnetic field 8, Ai can be conceived as the “vector potential” of the “magnetic field Bi” associated with the circulation of electrons round the flux lines. It 127
Volume 129, number 2
PHYSICS LETTERS A
9 May 1988
follows that the London field B, ( Bt = B+ Bi) has the value nh/e, where n is the 2D electron density, and Bi is related to the magnetization of the 2DES by $eAi*dL=
$m~dL.,
and for a simple ring geometry the result is Bi= (4rcm/e2)M.
(3)
Here M is defined as the magnetic moment per area. With the formation of a Cooper pair, the expression becomes Bi = ( 2xm/e2)M.
(5)
which is shown by the dashed line in the M-B diagram in fig. la. In fact, when f=BJB is an integer, frustration always occurs which induces Kosterlitz-Thouless type transitions, i.e. the pairing of electrons. Therefore between two frustration fields Bi is given by eq. (4) rather than by eq. (3 ), which shows diamagnetism, whereas the transitions are paramagnetic at frustration fields to recover back the singleelectron magnetization described by eq. (5) because frustrations are the consequences of single-electron Josephson arrays. It is clear that the magnetization is anisotropic with no spin splitting because only the orbital motion in the layer plane is considered. In this way we have an oscillation of the magnetization whose amplitude decreases as the applied magnetic field is decreased and shows no spin splitting as shown in fig. la by the solid line. The explicit dependence of magnetization oscillation amplitude on the Landau filling factor f is given in table 1 which is derived from fig. 1a. When the f are_rational fractions, corresponding to the fractional quantum Hall effect, frustrations are more complicated, as are the magnetization oscillations. In real 2DES, there are some factors preventing direct comparison of the theoretical magnetization 128
c (b)
1
(4)
Diamagnetism is a very important characteristic of a quantum 2DES, whether it is in the single-electron state or in the Cooper pair state, that is to say that the London flux through the layer is always constant. In the case of a single-electron state with no frustration, magnetization and applied magnetic field should have the linear relation ( 4Tcm/e2)M+ B= nh/e ,
-1 WE;,
1
B(nh/s) c
--1!? Fig. 1. (a) Orbital magnetization of 2DES versus applied magnetic field for integer Landau filling factor. (b) The upper magnetization is a sum of three terms: a constant term, a linear term, and an oscillation term. The number are the reciprocal Landau tilling factors. Here MOis equal to nefi/2m.
with the experimental one [ 1,2]. First at a non-zero temperature the discontinuities are smeared, so only continual oscillations could be observed generally. Second the electron mass m should be replaced by the effective mass m*, which is a function of temperature, magnetic field and layer mobility. In GaAs/AlGaAs heterostructures m* < m, which accounts for the large amplitudes observed in experiments. Third, background magnetizations arisen Table 1 Magnetization oscillation amplitude versus the Landau tilling factorj M (neh/2m)
f (nhleB)
1/(1X2) 1/(2x3) 1/(3X4) 1/(4X5) 1/(5x6) 1/(6x7) 1/(7x8)
1 2 3 4 5 6 7
.
suming that the relationship given in table 1 is not changed after a small smearing of the magnetization steps. The magnetic field at which the magnetization amplitude is a maximum must satisfy the condition that f= B,/B is an integer, where B,= nh/e, so the experimental Bt is calulated by l/B= 1/B,, and the Landau filling factor can be calculated consequently. The magnitude of one maximum amplitude must be known from experiment in order that a quantitative comparison between theory and experiment in all fields could be made, site only the relative amplitude has practical utility. The original and calculated parameters are listed in table 2. The original 2D parameters are taken from the work of Eisenstein et al.
.O.l
I 0
9 May 1988
PHYSICS LETTERS A
Volume 129, number 2
1
3
2
5
4
B(T)
111.
Fig. 2. Normalized magnetization oscillation amplitude versus applied magnetic field for all three samples: tilled circles, sample 1; squares, sample 2; open circles, sample 3. The solid lines are the theoretical envelopes described in the text. Notice that the point at the highest magnetic field in sample 2 has been excluded from the calculations because it does not satisfy the condition that B,/B is an integer. The numbers are the calculated Landau filling factors.
It can be seen that the calculated oscillation amplitudes agree with experiments quite well for all three samples and are considerably better than those assuming a density of states consisting of gaussian Landau levels [ 11. Magnetization oscillations are more complicated in ( TMTSF)&104 single crystals, provided that the electron density in a layer varies with magnetic field and temperature [ 2 1. However, paramagnetic jumps have been observed in them consistent with theory. All these contributed evidence for the microscopic picture of quantum 2DES in high magnetic fields and low temperatures described in ref. [ 51. Furthermore, in table 2, the quantity defined as the ratio of experimental Bt to theoretical B, is always smaller than unity, corresponding to the concept that the number of superconducting electrons in a system is always smaller than the the total number of electrons. In conclusion, a simple and elegant relationship has been derived between the oscillatory amplitudes of magnetization in a quantum 2DES in a magnetic field where the quantum Hall effect could be observed. The carrier orbital motion in a 2D perfect single electron Josephson lattice accounts for the an-
from electron spin and background atomic magnetism cause a smooth and a linear isotropic magnetization. Therefore, we must view the total magnetization as the sum of a smooth term, a linear term and an oscillatory term, as done in fig. lb. Despite all these factors, the general picture is not changed and the relative amplitude is limited to the relation given in table 1. Fig. 2 presents the experimental results obtained by Eisenstein et al. for three different samples. Here the amplitudes of the magneto-oscillations, normalized by the amplitude Mo=neii/2m*, are plotted versus the appied magnetic fields. These points provide a basis for comparison with the theoretical calculations. The solid lines represent the theoretical envelopes for the magnetization oscillations by asTable 2 Parameters for all three samples. Sample
(Wcxp (T)
(&&or
1 2 3
19.5 12.2 19.1
22.3 15.3 22.8
(Tl
WeXPl (&keor
f In,”
f
wI~oLl.x
0.83 0.80 0.84
4.7 5.7 3.6
5 6 4
0.214 0.133 0.163
129
Volume
129, number
2
PHYSICS
isotropic and paramagnetic jumps of the magnetooscillation in the ( TMTSF)$104 single cyrstal. I would like to thank Professors S.X. Zhou and X. Sun for helpful conversations.
References [ I] J.P. Eisenstein, H.L. Stormer, V. Narayanamurti, A.Y. Cho, A.C. Gossard and C.W. Tu, Phys. Rev. Lett. 55 (1985) 875.
130
LETTERS
A
9 May 1988
[2] M.J. Naughton, J.S. Brooks, L.Y. Chiang, R.V. Chamberlin and P.M. Chaikin, Phys. Rev. Lett. 55 (1985) 969. [3]P.M. Chaikin, M.-Y. Choi, L.F. Kwak, J.S. Brooks, K.P. Martin,M.J. Naughton, E.M. Engler and R.L. Green, Phys. Rev. Lett. 51 (1983) 2333. [4] D. Shoenberg, Magnetic oscillations in metals (Cambridge Univ. Press, Cambridge, 1984). [5] J.P. Peng, Phys. Lett. A 129 (1988) 124.
PHYSICS LETTERS A
9 May 1988
MAGNETIC FIELD DEPENDENT MAGNETIZATION OSCILLATIONS IN TWO-DIMENSIONAL QUANTUM ELECTRON SYSTEMS
J.P. PENG Department ofphysics, Fudan University, Shanghai, PR China
Received 13 July 1987; accepted for publication 1 March 1988 Communicated by A.A. Maradudin
A simple relationship between amplitudes of magnetization oscillations is presented as result of a single-electron Josephson junction in a two-dimensional quantum electron system with a perpendicular magnetic field.
Recent high-precision dc measurements of two-dimensional electron systems (2DES) in homogeneous high-mobility GaAs/AlGaAs single-layer and multilayer heterostructures [ 1 ] as well as in single crystals of ( TMTSF)&104 in the regime where the magnetic field induces a series of transitions show interesting pictures [ 21. Anisotropic oscillation of the magnetization has been observed whose amplitude attenuates with decreasing magnetic field and shows no observable spin splitting. In GaAs/AlGaAs heterostructures well-developed quantum Hall effects have been observed at low temperatures and in (TMTSF ) Z,ClO, single-electron crystals Shubnikov-de Haas (SdH ) type magnetoresistance oscillation as well as step-like Hall resistances have been observed [ 3 1. It is also worth mentioning that small, smooth isotropic and large anisotropic background magnetizations have been observed in these samples. Although the magnetoresistance and Hall data resemble SdH besides that the magnetization results resemble de Haas-van Alphen (dHvA) oscillations, there are strong reasons to believe that the dHvA oscillation gives not the entire picture of the magnetization. For a 2DES with fixed particle density, such oscillations at zero temperature appear as discontinuous diamagnetic (as the field is increased) jumps of magnitude &n/m* (field independent), where n is the zero-field carrier density, with a paramagnetic relaxation between jumps [ 41. Also in the regime where few Landau levels are occupied, there should be an observable spin splitting. For a fixed chemical
potential the jumps are paramagnetic and the recovery is diamagnetic and the curves will appear inverted. As the temperature is increased, the discontinuities are “smeared” but the magnetic fields at which they occur should remain the same. But even if only a small smearing fron the zero-temperature behavior is assumed, there are still major discrepancies related to the increasing amplitude with applied magnetic field, the apparent absence of spin splitting in the case of (TMTSF) &lo, single crystal. In this Letter a quantitative description is presented for the oscillatory amplitude attenuation with increasing applied magnetic field in quantum 2DES as well as a comparison with recent experiment. Motivated by the concept of frustration in two-dimensional superconducting arrays, a theory of the quantum Hall effect has been given in the preceding Letter [ 51 with the hypothesis of the single-electron Josephson arrays where quantization of the London flux and the formation of Cooper pairs are essential [ 5 1. The London quantization condition in a cell is s
(mv/e+A)dL=h/e,
(1)
which can be written as
s
(Ai+A)*dL=h/e,
(2)
where A is the vector potential of the applied magnetic field 8, Ai can be conceived as the “vector potential” of the “magnetic field Bi” associated with the circulation of electrons round the flux lines. It 127
Volume 129, number 2
PHYSICS LETTERS A
9 May 1988
follows that the London field B, ( Bt = B+ Bi) has the value nh/e, where n is the 2D electron density, and Bi is related to the magnetization of the 2DES by $eAi*dL=
$m~dL.,
and for a simple ring geometry the result is Bi= (4rcm/e2)M.
(3)
Here M is defined as the magnetic moment per area. With the formation of a Cooper pair, the expression becomes Bi = ( 2xm/e2)M.
(5)
which is shown by the dashed line in the M-B diagram in fig. la. In fact, when f=BJB is an integer, frustration always occurs which induces Kosterlitz-Thouless type transitions, i.e. the pairing of electrons. Therefore between two frustration fields Bi is given by eq. (4) rather than by eq. (3 ), which shows diamagnetism, whereas the transitions are paramagnetic at frustration fields to recover back the singleelectron magnetization described by eq. (5) because frustrations are the consequences of single-electron Josephson arrays. It is clear that the magnetization is anisotropic with no spin splitting because only the orbital motion in the layer plane is considered. In this way we have an oscillation of the magnetization whose amplitude decreases as the applied magnetic field is decreased and shows no spin splitting as shown in fig. la by the solid line. The explicit dependence of magnetization oscillation amplitude on the Landau filling factor f is given in table 1 which is derived from fig. 1a. When the f are_rational fractions, corresponding to the fractional quantum Hall effect, frustrations are more complicated, as are the magnetization oscillations. In real 2DES, there are some factors preventing direct comparison of the theoretical magnetization 128
c (b)
1
(4)
Diamagnetism is a very important characteristic of a quantum 2DES, whether it is in the single-electron state or in the Cooper pair state, that is to say that the London flux through the layer is always constant. In the case of a single-electron state with no frustration, magnetization and applied magnetic field should have the linear relation ( 4Tcm/e2)M+ B= nh/e ,
-1 WE;,
1
B(nh/s) c
--1!? Fig. 1. (a) Orbital magnetization of 2DES versus applied magnetic field for integer Landau filling factor. (b) The upper magnetization is a sum of three terms: a constant term, a linear term, and an oscillation term. The number are the reciprocal Landau tilling factors. Here MOis equal to nefi/2m.
with the experimental one [ 1,2]. First at a non-zero temperature the discontinuities are smeared, so only continual oscillations could be observed generally. Second the electron mass m should be replaced by the effective mass m*, which is a function of temperature, magnetic field and layer mobility. In GaAs/AlGaAs heterostructures m* < m, which accounts for the large amplitudes observed in experiments. Third, background magnetizations arisen Table 1 Magnetization oscillation amplitude versus the Landau tilling factorj M (neh/2m)
f (nhleB)
1/(1X2) 1/(2x3) 1/(3X4) 1/(4X5) 1/(5x6) 1/(6x7) 1/(7x8)
1 2 3 4 5 6 7
.
suming that the relationship given in table 1 is not changed after a small smearing of the magnetization steps. The magnetic field at which the magnetization amplitude is a maximum must satisfy the condition that f= B,/B is an integer, where B,= nh/e, so the experimental Bt is calulated by l/B= 1/B,, and the Landau filling factor can be calculated consequently. The magnitude of one maximum amplitude must be known from experiment in order that a quantitative comparison between theory and experiment in all fields could be made, site only the relative amplitude has practical utility. The original and calculated parameters are listed in table 2. The original 2D parameters are taken from the work of Eisenstein et al.
.O.l
I 0
9 May 1988
PHYSICS LETTERS A
Volume 129, number 2
1
3
2
5
4
B(T)
111.
Fig. 2. Normalized magnetization oscillation amplitude versus applied magnetic field for all three samples: tilled circles, sample 1; squares, sample 2; open circles, sample 3. The solid lines are the theoretical envelopes described in the text. Notice that the point at the highest magnetic field in sample 2 has been excluded from the calculations because it does not satisfy the condition that B,/B is an integer. The numbers are the calculated Landau filling factors.
It can be seen that the calculated oscillation amplitudes agree with experiments quite well for all three samples and are considerably better than those assuming a density of states consisting of gaussian Landau levels [ 11. Magnetization oscillations are more complicated in ( TMTSF)&104 single crystals, provided that the electron density in a layer varies with magnetic field and temperature [ 2 1. However, paramagnetic jumps have been observed in them consistent with theory. All these contributed evidence for the microscopic picture of quantum 2DES in high magnetic fields and low temperatures described in ref. [ 51. Furthermore, in table 2, the quantity defined as the ratio of experimental Bt to theoretical B, is always smaller than unity, corresponding to the concept that the number of superconducting electrons in a system is always smaller than the the total number of electrons. In conclusion, a simple and elegant relationship has been derived between the oscillatory amplitudes of magnetization in a quantum 2DES in a magnetic field where the quantum Hall effect could be observed. The carrier orbital motion in a 2D perfect single electron Josephson lattice accounts for the an-
from electron spin and background atomic magnetism cause a smooth and a linear isotropic magnetization. Therefore, we must view the total magnetization as the sum of a smooth term, a linear term and an oscillatory term, as done in fig. lb. Despite all these factors, the general picture is not changed and the relative amplitude is limited to the relation given in table 1. Fig. 2 presents the experimental results obtained by Eisenstein et al. for three different samples. Here the amplitudes of the magneto-oscillations, normalized by the amplitude Mo=neii/2m*, are plotted versus the appied magnetic fields. These points provide a basis for comparison with the theoretical calculations. The solid lines represent the theoretical envelopes for the magnetization oscillations by asTable 2 Parameters for all three samples. Sample
(Wcxp (T)
(&&or
1 2 3
19.5 12.2 19.1
22.3 15.3 22.8
(Tl
WeXPl (&keor
f In,”
f
wI~oLl.x
0.83 0.80 0.84
4.7 5.7 3.6
5 6 4
0.214 0.133 0.163
129
Volume
129, number
2
PHYSICS
isotropic and paramagnetic jumps of the magnetooscillation in the ( TMTSF)$104 single cyrstal. I would like to thank Professors S.X. Zhou and X. Sun for helpful conversations.
References [ I] J.P. Eisenstein, H.L. Stormer, V. Narayanamurti, A.Y. Cho, A.C. Gossard and C.W. Tu, Phys. Rev. Lett. 55 (1985) 875.
130
LETTERS
A
9 May 1988
[2] M.J. Naughton, J.S. Brooks, L.Y. Chiang, R.V. Chamberlin and P.M. Chaikin, Phys. Rev. Lett. 55 (1985) 969. [3]P.M. Chaikin, M.-Y. Choi, L.F. Kwak, J.S. Brooks, K.P. Martin,M.J. Naughton, E.M. Engler and R.L. Green, Phys. Rev. Lett. 51 (1983) 2333. [4] D. Shoenberg, Magnetic oscillations in metals (Cambridge Univ. Press, Cambridge, 1984). [5] J.P. Peng, Phys. Lett. A 129 (1988) 124.